DPNU-01-05 hep-lat/0105032
arXiv:hep-lat/0105032v2 15 Jun 2001
Domain wall fermion and chiral gauge theories on the lattice with exact gauge invariance Yoshio Kikukawa∗ Department of Physics, Nagoya University Nagoya 464-8602, Japan
May, 2001
Abstract We discuss how to construct anomaly-free chiral gauge theories on the lattice with exact gauge invariance in the framework of domain wall fermion. Chiral gauge coupling is realized by introducing a five-dimensional gauge field which interpolates between two different four-dimensional gauge fields at boundaries. The five-dimensional dependence is compensated by a local and gauge-invariant counter term. The cohomology problem to obtain the counter term is formulated in 5+1 dimensional space, using the Chern-Simons current induced from the five-dimensional Wilson fermion. We clarify the connection to the invariant construction based on the Ginsparg-Wilson relation using overlap Dirac operator. Formula for the measure and the effective action of Weyl fermions are obtained in terms of five-dimensional lattice quantities.
∗
e-mail address:
[email protected]
1
1
Introduction
Through the Ginsparg-Wilson relation [1] and the exact chiral symmetry based on it [2], Weyl fermion can naturally be introduced on the lattice. The chiral constraint imposed on the Weyl fermion is gauge-field dependent and by introducing the basis of Weyl fermion, the path integral can be set up [3, 4]. In fact, it has been shown by L¨ uscher that the functional measure of the Weyl fermion can be constructed in anomaly-free abelian chiral gauge theories so that it satisfies the requirements of the smoothness, the locality and the gauge invariance [3, 5, 6, 7]. The similar construction has also been argued for generic non-abelian chiral gauge theories, where to treat the exact cancellation of gauge anomaly, a local cohomology problem in 4 + 2-dimensions is formulated [4].1 This construction is generic and applies to any local lattice fermion theory with the Dirac operator satisfying the Ginsparg-Wilson relation. In the case using the overlap Dirac operator [14, 15, 16], the path integral formalism for the Weyl fermion reproduces the overlap formula for the chiral determinant by Narayanan and Neuberger [17].2 The above invariant construction of the functional measure provides the method to fix the phase factor of the chiral determinant in the overlap formalism in a gauge-invariant manner. It was also suggested [4, 28] that there is a close relation between the interpolation procedure in L¨ uscher’s construction and the five-dimensional setup in Kaplan’s domain wall fermion [29]. The purpose of this paper is to pursue this close connection and to show how to construct four-dimensional lattice chiral gauge theories with exact gauge invariance from the five- di1 The author refers the reader to [8, 9] for recent review of this approach. In this approach, the exact cancellation of gauge anomalies in non-abelian chiral gauge theories has been shown in all orders of the expansion in lattice perturbation theory [10, 11]. For SU(2) doublet, it has been shown that Witten’s global anomaly is reproduced [12]. For SU(2)L × U(1)Y electroweak theory, the local cohomology problem in 4 + 2-dimensions has be solved in infinite volume lattice and the exact cancellation of gauge anomalies, including the mixed type, has been shown non-perturbatively [13]. 2 The author refers the reader to [18] for recent review of the overlap formalism. In the overlap formalism, reflecting chiral anomaly, the phase factor of the chiral determinant is not fixed in general and any reasonable choice of the phase factor should lead to the gauge anomaly for single Weyl fermion. The Wigner-Brillouin phase convention has been adopted for perturbative studies [19] and has also been tested numerically in a non-perturbative formulation of chiral gauge theories [20, 21]. Geometrical treatment of the gauge anomaly in the overlap formalism has been discussed in detail in abelian theories [22] and non-abelian theories [23]. The SU(2) global anomaly has been examined in [24]. An adiabatic phase choice has been proposed in [25] and used in the construction of non-compact abelian chiral gauge theories. The overlap formalism in odd dimensions has been considered in [26, 16, 27].
2
mensional lattice framework of domain wall fermion. For this purpose, we adopt the domain wall fermion in the vector-like formalism by Shamir [30]. But two different four-dimensional gauge fields are introduced at the boundaries and they are interpolated by a five-dimensional gauge field. This inevitably causes the five-dimensional dependence of the partition function of the domain wall fermion. To take account of this five-dimensional dependence, we formulate an integrability condition. It turns out that the dependence is governed by the lattice Chern-Simons term induced from the five-dimensional Wilson-Dirac fermion (with a negative mass) [31]. In order to compensate the five-dimensional dependence, we require a five-dimensional counter term. The counter term should be given by a smooth, local and gauge invariant functional of gauge field, in order to satisfy the requirement of the smoothness, the locality and the gauge invariance of the low energy effective action. We will argue that such local, gauge-invariant field can be obtained in anomaly-free chiral gauge theories, through the local cohomology problem in 5 + 1-dimensional space formulated with the lattice Chern-Simons current. Thus the reduction from the five-dimensional lattice to four-dimensional lattice is acheived in a local and gauge invariant manner. The locality of the lattice Chern-Simons current is essential for the cohomological argument in 5 + 1-dimensional space and for this we require the so-called admissibility condition [32, 5, 3, 4] extended to five-dimensional gauge fields (cf. [34]). With this condition, several properties of the ChernSimons current are discussed. The earlier studies of the properties of the Chern-Simons current in the context of domain wall fermion can be found in [35, 36]. Trying to formulate four-dimensional chiral gauge theories from the fivedimensional framework of domain wall fermion, our approach resembles to the wave-guide model [37, 38] and the formalism proposed by Creutz et al. [39, 40]. However, our approach is different from the wave-guide model in that we are considering the smooth (discrete, but smooth in lattice scale) interpolation in the fifth direction. The issue related to the disordered gauge degrees of freedom is taken account by the five-dimensional admissibility condition, which assures the existence of the chiral zero modes even with five-dimensional gauge fields (cf. [41, 42, 43]). Our approach is also different from the formalism by Creutz et al. in that we are isolating the chiral zero modes at one boundary as physical degrees of freedom, regarding the other boundary as reference. This paper is organized as follows. In section 2 we formulate the domain 3
wall fermion for chiral gauge theories with the interpolating five-dimensional gauge field. Then we derive the integrability condition for the partition function of the domain wall fermion and state a sufficient condition to obtain the five-dimensional counter term with the required properties. In section 3 we examine the properties of the lattice Chern-Simons current. In section 4 we argue how to reconstruct the counter term from the Chern-Simons current and formulate the local cohomology problem in 5+1 dimensional space. Section 5 is devoted to the discussions on the connection to the gauge-invariant construction based on the Ginsparg-Wilson relation.
2 2.1
Domain wall fermion for chiral gauge theories Interpolation with five-dimensional gauge field
Domain wall fermion, in its simpler vector-like formulation, is defined by the five-dimensional Wilson-Dirac fermion with a negative mass in a finite extent fifth dimension. (See Figure 1.) The four-dimensional lattice spacing a and the five-dimensional one a5 are both set to unity. The fifth coordinate is denoted by t and takes integer values in the interval, t ∈ [−N + 1, N ]. In four dimensions, the lattice is assumed to have a finite volume L4 and the periodic boundary condition is assumed for both fermion and gauge fields. Mass term is set to the negative value −m0 where 0 < m0 < 2. SDW =
N X
X
t=−N +1 x
¯ t) (D5w − m0 ) ψ(x, t). ψ(x,
(2.1)
This setup is equivalent to impose the Dirichlet boundary condition at the boundaries in the fifth dimension as ψR (x, t)|t=−N = 0,
ψL (x, t)|t=N +1 = 0.
(2.2)
In order to introduce chiral-asymmetric gauge interaction for the chiral zero modes at the two boundaries t = −N + 1 and N , the gauge field is assumed to be five-dimensional, Uµ (z) = {Uk (x, t), U5 (x, t)} ,
z = (x, t)
(2.3)
where µ = 1, · · · , 5 and k = 1, · · · , 4. It is regarded to be interpolating a four-dimensional gauge field at t = −N + 1, say Uk0 (x), to another fourdimensional gauge field at t = N , say Uk1 (x). We assume that outside the finite interpolation region t ∈ [−∆, ∆](∆ < N ) the gauge field does not 4
(+m0 )
ΨL (x)
−m0
ΨR (x)
(+m0 )
t = −N + 1
- x = ta 5 5
t=N
Figure 1: Domain wall fermion in the simpler vector-like setup
depend on t and U5 (x, t) = 1 (Figure 2). ∆ should be chosen large enough in order to make sure that the interpolation is smooth enough. The precise condition for this will be discussed below.
Uk1 (x) Uµ (x, t) Uk0 (x) r
−N + 1
r
r
r
r
r
r
-∆
r
r
r
0
r
r
r
r
r
r
r
∆
r
N
Figure 2: Interpolating five-dimensional gauge field on the lattice The gauge fields at the boundaries, Uk0 (x) and Uk1 (x), are chosen so that their field strengths are small enough and satisfy the following bound k1 − Pkl (x)k < ǫ,
ǫ
